However, there's conflicting information regarding the Braves, as Mark Bowman of MLB.com is reporting the team hasn't made any offers to Kimbrel. In addition, a report earlier this month suggested the team lost interest.
The Brewers, meanwhile, are also facing bullpen health issues.
Corey Knebel will undergo an examination of his right elbow after manager Craig Counsell revealed Thursday that the right-hander has an ulnar collateral ligament issue. Milwaukee is also expected to start the season without closer Jeremy Jeffress (sore shoulder), and Bobby Wahl is done for the year after ACL surgery.
The Tampa Bay Rays have also reportedly maintained interest in Kimbrel, while the Washington Nationals appear to be out of the running.
The seven-time All-Star is attached to draft-pick compensation after turning down a qualifying offer from the Boston Red Sox following the 2018 campaign.
The 30-year-old has been a force over nine MLB seasons spent with the Braves, Red Sox, and San Diego Padres. Kimbrel has recorded 333 career saves with a sparkling 1.91 ERA and 14.7 K/9 rate.